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Determinantal characterization of canonical curves and combinatorial theta identities

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Abstract

We characterize genus g canonical curves by the vanishing of combinatorial products of g + 1 determinants of Brill-Noether matrices. This also implies the characterization of canonical curves in terms of (g−2)(g−3)/2 theta identities. A remarkable mechanism, based on a basis of H 0(K C ) expressed in terms of Szegö kernels, reduces such identities to a simple rank condition for matrices whose entries are logarithmic derivatives of theta functions. Such a basis, together with the Fay trisecant identity, also leads to the solution of the question of expressing the determinant of Brill-Noether matrices in terms of theta functions, without using the problematic Klein-Fay section σ.

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Author notes

  1. Roberto Volpato

    Present address: Max-Planck-Institut für Gravitationsphysik, Am Mühlenberg, 1, 14476, Potsdam, Golm, Germany

Authors and Affiliations

  1. Dipartimento di Fisica “G. Galilei” and Istituto Nazionale di Fisica Nucleare, Università di Padova, Via Marzolo, 8, 35131, Padua, Italy

    Marco Matone & Roberto Volpato

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  1. Marco Matone
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  2. Roberto Volpato
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Correspondence to Marco Matone.

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Matone, M., Volpato, R. Determinantal characterization of canonical curves and combinatorial theta identities. Math. Ann. 355, 327–362 (2013). https://doi.org/10.1007/s00208-012-0787-z

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  • DOI: https://doi.org/10.1007/s00208-012-0787-z

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